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Beliefs and Pareto Efficient Sets: A Remark
Thibault Gajdos, Jean-Marc Tallon
To cite this version:
Thibault Gajdos, Jean-Marc Tallon. Beliefs and Pareto Efficient Sets: A Remark. Journal of Economic Theory, Elsevier, 2002, 106 (2), pp.467-471. �10.1006/jeth.2001.2899�. �halshs-00085912�
Beliefs and Pareto Efficient Sets: a Remark
∗Thibault Gajdos†
Universit´e Cergy-Pontoise, THEMA,
&
EUREQua, Universit´e Paris I,
106-112 bld de l’Hˆopital,
75647 Paris Cedex 13, France. e-mail: gajdos@univ-paris1.fr
Jean-Marc Tallon
EUREQua, CNRS-Universit´e Paris I,
106-112 bld de l’Hˆopital,
75647 Paris Cedex 13, France &
Universita Ca’Foscari, Venezia, Italy. March 2001
∗
We thank E. Dekel, I. Gilboa, P. Gourdel, Z. Safra, and D. Schmeidler for useful comments and discussions. We are grateful to an anonymous referee for pointing out a mistake in an earlier draft.
†
Abstract
We show that, in a two-period economy with uncertainty in the second period, if an allocation is Pareto optimal for a given set of beliefs and remains optimal when these beliefs are changed, then the set of optimal allocations of the two economies must actually coincide. We identify equivalence classes of beliefs giving rise to the same set of Pareto optimal allocations.
Journal of Economic Literature Classification Numbers: D51, D61.
Key words: Beliefs, Pareto Optimality.
1
Introduction
In this note, we seek to answer a very simple question: what can we learn about agents’ beliefs by the sole knowledge that a given allocation is Pareto optimal? More specifically, consider a multiple-goods, two-period economy with uncertainty in the second period and agents that are subjective expected utility maximizers. Take a Pareto optimal allocation of this economy. Is it possible that this allocation still be Pareto optimal in an economy in which agents’ beliefs have changed?
We answer this question affirmatively and actually identify the exact change of beliefs needed. The result we obtain is actually stronger: if agent h’s subjective probability of state s divided by that of state s0 in the second economy (i.e., the economy after beliefs have changed) is proportional (with the same coefficient of proportionality for all the agents) to the same ratio in the initial economy, then, the set of Pareto optimal allocations is the same in those two economies. We furthermore show that this is equivalent to the two sets of Pareto optimal allocations having one (interior) point in common. Hence, two contract curves associated to two economies with different beliefs are either equal or disjoint.
To the best of our knowledge, this point, as simple as it seems, has not been studied in the literature. In a sense, the class of probabilities we identify is similar to what Radner (1979) called “confounding” probabilities in a (rational expectations) equilibrium set-up, since in our set-up, two such sets of beliefs lead to the same Pareto optimal set.
2
The set up and main result
We consider a standard two period economy with uncertainty in the second period. There are H agents, h = 1, ..., H and C commodities, c = 1, ..., C, in each spot market. Without loss of generality we assume that there is no consumption in the first period. Uncertainty is represented by a state space S = {1, ..., S}, with s ∈ S a state of nature. Total contingent endowments are given by e = (e (1) , ..., e (S)) ∈ RCS++.
Agents are subjective expected utility maximizers with beliefs πh = (πh(1) , ..., πh(S)). It is
assumed that πh(s) > 0 ∀s ∈ {1, ..., S} and, naturally that
P
sπh(s) = 1 for all h. Agent h
has consumption set RCS++, certainty preferences represented by the von Neumann-Morgenstern
utility index uh : RC++ → R. uh is assumed to be twice continuously differentiable,
differ-entiably strictly increasing (i.e., ∇uh(x) 0 for x 0) and differentiably strictly concave
(i.e., ∆xt∇2u
h(x) ∆x < 0 for x 0, ∆x 6= 0), and to have indifference surfaces with closures
in RC++. Finally, the household evaluates its contingent consumption plan, represented by the
Vh(xh(1) , ..., xh(S)) =Psπh(s) uh(xh(s)).
An allocation x = (x1, ..., xH) is feasible if xh(s) 0 for all h and all s and PHh=1xh(s) =
e (s) for all s. An allocation x is Pareto optimal if there is no other feasible allocation x0 such that Vh(x0h) ≥ Vh(xh) for all h and Vh(x0h) > Vh(xh) for some h.
In this note, we take the von Neumann-Morgenstern utility indices and total endowments to be fixed and allow changes in agents’ beliefs. Let P (π) be the set of Pareto optima of the economy where agents have beliefs π = (π1, ..., πH). Recall that in our simple set-up1 an allocation x is a
Pareto optimal allocation if and only if there exists a vector of weights λ = (λ1, ..., λH) 0 such
that x is a solution to the problem maxPH
h=1λhPSs=1πh(s) uh(xh(s)) s.t. PHh=1xh(s) = e (s)
for all s and xh 0 for all h.
The main result of this note is to compare the set of Pareto optimal allocations in two economies differing only by the agents’ beliefs.
Proposition 1 The following three assertions are equivalent: (i) P (π) = P (ˆπ) (ii) P (π) ∩ P (ˆπ) 6= ∅ (iii) ∀h, h0, ∀s, s0, πh(s)/πh(s0) πh0(s)/πh0(s0) = ˆ πh(s)/ˆπh(s0) ˆ πh0(s)/ˆπh0(s0) Proof.
Recall first the following lemma (see, e.g., Cass, Chichilnisky, and Wu (1996)):
Lemma: A feasible allocation x is Pareto optimal if and only if there exist positive weights, λh> 0, all h, and strictly positive contingent goods prices (multipliers) for each state, µ(s) 0,
all s, such that
λhπh(s) ∇uh(xh(s)) = µ(s), all h, s
Let us now prove our result. That (i) implies (ii) is trivial. (ii) =⇒ (iii)
Assume that P (π) ∩ P (ˆπ) 6= ∅ and pick a feasible allocation x in P (π) ∩ P (ˆπ) . Then, there exist λ = (λ1, ..., λH) 0 and ˆλ = ˆλ1, ..., ˆλH
0 as well as µ = (µ(1), ..., µ(S)) and b
µ = (µ(1), ...,b µ(S)) such that, for all h, hb 0 and all s:
λhπh(s) ∇uh(xh(s)) = λh0πh0(s) ∇uh0(xh0(s)) = µ(s)
ˆ
λhπˆh(s) ∇uh(xh(s)) = ˆλh0ˆπh0(s) ∇uh0(xh0(s)) =
b µ(s)
Hence, λhπh(s) λh0πh0(s) = ˆ λhˆπh(s) ˆ λh0ˆπh0(s) , ∀h, h0, s
Therefore, for all s, s0, h and h0: πh(s) πh0(s) ˆ πh0(s) ˆ πh(s) = πh(s 0) πh0(s0) ˆ πh0(s0) ˆ πh(s0) proving (iii). (iii) =⇒ (i)
Let x ∈ P (π). Then, by the lemma, there exists a vector of weights λ = (λ1, ..., λH) 0
and multipliers (contingent goods prices) µ = (µ(1), ..., µ(S)) ∈ RCS++ such that, for all h and all
s: λhπh(s) ∇uh(xh(s)) = µ(s) (1) Now, by assumption, πh(s) /πh(1) π1(s) /π1(1) = πˆh(s) /ˆπh(1) ˆ π1(s) /ˆπ1(1)
for all h and all s. Hence, (1) is equivalent to: λh πh(1) b πh(1) ˆ πh(s) ∇uh(xh(s)) = π1(1) π1(s) b π1(s) b π1(1) µ(s) for all h and all s. Therefore, defining bλh = λhπbπh(1)h(1) and µ(s) =b
π1(1) π1(s) bπ1(s) bπ1(1)µ(s), we get that ˆ λhπˆh(s) ∇uh(xh(s)) =µ(s)b
for all h and all s. Since bλh > 0 and µ(s) > 0, this establishes (by the lemma above) that xb ∈ P (ˆπ). Therefore, P (π) ⊆ P (ˆπ). The converse inclusion also holds by a symmetric argument. Hence P (π) = P (ˆπ).
Observe that condition (iii) in the proposition does not imply that πh = ˆπh for all h, as
shown by the following example: H = 2, S = 2, and π1(1) = 14, π2(1) = 13, ˆπ1(1) = 1316 and
ˆ
π2(1) = 1315.
To interpret condition (iii), observe that the ratio πh(sπh(s)0) is simply the marginal rate of
substitution, say of good 1, between state s and s0 when agent h is risk neutral (i.e., has a linear utility index). Alternatively, it is the marginal rate of substitution between state s and s0 at points where the consumer is fully insured, i.e., consumes the same bundle in each of these states.
Remark 1 If we were to take RCS+ rather than RCS++ as households’ consumption set and extend
the domain of the utility function accordingly, the same result would continue to hold, in which
Remark 2 The framework developed can be reinterpreted in an intertemporal setting, with time-separable, time-independent preferences. Indeed, interpreting s as a time index and writing πh(s) = (βh)s where βh is h’s stationary discount factor, our result says that if the discount
factor changes but the ratios of the discount factors for any two agents remain the same, then the two economies have the same Pareto optima.
References
Cass, D., G. Chichilnisky, and H.-M. Wu (1996): “Individual risk and mutual insurance,” Econometrica, 64(2), 333–341.
Radner, R. (1979): “Rational expectations equilibrium : Generic existence and the information revealed by prices,” Econometrica, 47, 655–678.
